Assume that a random sample is used to

estimate a population proportion p. Find
the margin of error E corresponds to the given statistics and confidence level.
90% confidence, the sample size is 1457,
of which 30% are successes.

To find the margin of error (E) for estimating a population proportion, we can use the formula:

E = z * sqrt((p̂ * q̂) / n)

Where:
- z is the z-score corresponding to the desired confidence level
- p̂ is the sample proportion (expressed as a decimal) obtained from the sample
- q̂ is the complement of the sample proportion, calculated as 1 - p̂
- n is the sample size

In this case, the given information is:
- Confidence level: 90% (which means α = 1 - 0.9 = 0.1)
- Sample size (n): 1457
- Sample proportion (p̂): 30% (which can be expressed as 0.30)

First, let's find the complement of the sample proportion:
q̂ = 1 - p̂
q̂ = 1 - 0.30
q̂ = 0.70

Next, we need to find the z-score corresponding to a 90% confidence level. The z-score can be obtained from the standard normal distribution table or calculator. For a 90% confidence level, z ≈ 1.645.

Now, let's calculate the margin of error (E):
E = z * sqrt((p̂ * q̂) / n)
E = 1.645 * sqrt((0.30 * 0.70) / 1457)

Calculating this expression will give you the margin of error (E) for estimating the population proportion with a 90% confidence level using the given sample statistics.